A couple of posts ago, I mentioned Max Jammer’s book “Concepts of Space” as a nice genealogy of that concept, with one shortcoming from my point of view – namely, as the subtitle suggests, it’s a “History of Theories of Space in Physics”, and since physics tends to use concepts out of mathematics, it lags a bit – at least as regards fundamental concepts. Riemannian geometry predates Einstein’s use of it in General Relativity by fifty some years, for example. Heisenberg reinvented matrices and matrix multiplication (which eventually led to wholesale importation of group theory and representation theory into physics). More examples no doubt could be found (String Theory purports to be a counterexample, though opinions differ as to whether it is real physics, or “merely” important mathematics; until it starts interacting with experiments, I’m inclined to the latter, though of course contra Hardy, all important mathematics eventually becomes useful for something).

What I said was that it would be nice to see further investigation of concepts of space within mathematics, in particular Grothendieck’s and Connes’. Well, in a different context I was referred to this survey paper by Pierre Cartier from a few years back, “A Mad Day’s Work: From Grothendieck To Connes And Kontsevich, The Evolution Of Concepts Of Space And Symmetry”, which does at least some of that – it’s a fairly big-picture review that touches on the relationship between these new ideas of space. It follows that stream of the story of space up to the end of the 20th century or so.

There’s also a little historical/biographical note on Alexander Grothendieck – the historical context is nice to see (one of the appealing things about Jammer’s book). In this case, much of the interesting detail is more relevant if you find recent European political history interesting – but I do, so that’s okay. In fact, I think it’s helpful – maybe not mathematically, but in other ways – to understand the development of mathematical ideas in the context of history. This view seems to be better received the more ancient the history in question.

On the scientific end, Cartier tries to explain Grothendieck’s point of view of space – in particular what we now call  topos theory – and how it developed, as well as how it relates to Connes’.  Pleasantly enough, a key link between them turns out to be groupoids!  However, I’ll pass on commenting on that at the moment.

Instead, let me take a bit of a tangent and jump back to Jammer’s book.  I’ll tell you something from his chapter “Emancipation from Aristotelianism” which I found intriguing.  This would be an atomistic theory of space – an idea that’s now beginning to make something of a comeback, in the guise of some of the efforts toward a quantum theory of gravity (EDIT: but see comments below).  Loop quantum gravity, for example, deals with space in terms of observables, which happen to take the form of holonomies of connections around loops.  Some of these observables have interpretations in terms of lengths, areas, and volumes.  It’s a prediction of LQG that these measurements should have “quantized”, which is to say integer, values: states of LQG are “spin networks”, which is to say graphs with (quantized) labels on the edges, interpreted as areas (in a dual cell complex).  (Notice this is yet again another, different, view of space, different from Grothendieck’s or Connes’, but shares with Connes especially the idea of probing space in some empirical way.  Grothendieck “probes” space mainly via cohomology – how “empirical” that is depends on your point of view.)

The atomistic theory of space Jammer talks about is very different, but it does also come from trying to reconcile a discrete “quantum” theory of matter with a theory linking matter to space.  In particular, the medieval Muslim philosophical school known as al Kalam tried to reconcile the Koran and Islamic theology with Greek philosophy (most of the “Hellenistic” world conquered by Alexander the Great, not least Egypt, is inside Dar al Islam, which is why many important Greek texts came into Europe via Arabic translations).  Though they were, as Jammer says, “Emancipating” themselves from Aristotle, they did share some of his ideas about space.

For Aristotle, space meant “place” – the answer to the questions “where is it?” and “what is its shape and size?”. In particular, it was first and foremost an attribute of some substance.  All “where?” questions are about some THING.  The answer is defined in terms of other things: my cat is on the ground, under the tree, beside the house.  The “place” of an object was literally the inner shell of the containing body that held it (which was contained by some other body, and so on – there being no vacuum in Aristotle).  So my “place” is defined by (depending how you look at it) my skin, my clothes, or the walls of the room I’m in.  This is a relational view of space, though more hard-headed than, say, Leibniz’s.

The philosophers of the Kalam had a similar relational view of space, but they didn’t accept Aristotle’s view of “substances”, where each thing has its own essential identity, on which attributes are hung like hats.  Instead, they believed in atomism, following Democritus and Leucippus: bodies were made out of little indivisible nuggets called “atoms”.  Macroscopic things were composites of atoms, and their attributes resulted from how the atoms were put together.  Here’s Jammer’s description:

The atoms of the Kalam are indivisible particles, equal to each other and devoid of all extension.  Spatial magnitude can be attributed only to a combination of atoms forming a body.  Although a definite position (hayyiz) belongs to each individual atom, it does not occupy space (makan).  It is rather the set of these positions – one is almost tempted to say, the system of relations – that constitutes spatial extension….

In the Kalam, these rather complicated and surprisingly abstract ideas were deemed necessary in order to meet Aristotle’s objections against atomism on the ground that a spatial continuum cannot be constituted by, or resolved into, indivisibles nor can two points be continuous or contiguous with one another.

So like people who prefer a “background independent” quantum theory of gravity, they wanted to believe that space (geometry) derives from matter, and that matter is discrete, but space was commonly held to be continuous.  Also alike, they resolved the problem by discarding the assumption of continuous space, and, by consideration of motion, to discrete time.

There are some differences, though.  The most obvious is that the nodes of the graph in a spin network state don’t represent units of matter, or “atoms”.  For that matter, quantum field theory doesn’t really have “atoms” in the sense of indivisible units which don’t break apart or interact.  Everything interacts in QFT.  (In some sense, interactions are more fundamental units in QFT than “particles” are – particles only (sic!) serve to connect one interaction with another.)

Another key difference is how space relates to matter.  In Aristotle, and in the Kalam, space is defined directly by matter: two bits of matter “define” the space between them.  In General Relativity (the modern theory with the “relational” view of space), there’s still room for space as an actor in its own right, like Newton’s absolute space-as-independent-variable – in other words, room for a vacuum, which Aristotle categorically denied could even conceivably exist.  In GR, what matter determines is the curvature of space (more precisely the Einstein tensor of the curvature).

Well, so the differences are probably more informative than the similarities,

(Edit: To emphasize a key difference glossed over before…  It was coupling to quantum matter which suggested quantizing the picture of space.  Discreteness of the spectrum of various observables is a logically separate prediction in each case.  Either matter or space(time) could have had continuous spectrum for the relevant observables and still been quantized – discrete matter would have given discreteness for some observed quantities, but not area, length, and so on.  So in the modern setting, the link is much less direct.)

 but the fact that theories of related discreteness in matter, space, and time, have been around for a thousand years or more is intriguing.  The idea of empty space as an independent entity – in the modern form only about three hundred years old – appears to be the real novel part.  One of the nice intuitions in Carlo Rovelli’s book on Quantum Gravity, for me at least, was to say that, rather than there being a separate “space”, we have a theory of fields defined on other fields as background – one of which, the “gravitational field” has customarily been taken for “space”.  So spatial geometry is a field, and it has some propagating (through space!) degrees of freedom – the particle associated to this field is a graviton.  Nobody’s ever seen one, mind you – but supposing they exist makes many of things easier.

To re-state a previous point: I think this is a nice aspect of categorification for dealing with space.  Extending the “stuff/structure/properties” trichotomy to allow space to resemble both “stuff” and relations between stuff leaves room for both points of view.

I mention this because tomorrow I leave London (Ontario) for London (England), and thence to Nottingham, for the Quantum Gravity and Quantum Geometry Conference.  It’s been a while since I worked much on quantum gravity, per se, but this conference should be interesting because it seems to be a confluence of mathematically and physically inclined people, as the name suggests.  I read on the program, for example, that Jerzy Lewandowski is speaking on QFT in Quantum Curved Spacetime, and suddenly remember that, oh yes, I did a Masters thesis (viz) on QFT in curved (classical) spacetime… but that was back in the 20th century!

It’s been a while, and I only made a small start at it before, but that whole area of physics is quite pretty.  Anyway, it should be interesting, and there are a number of people I’m looking forward to talking to.